An introduction to Homotopy Type Theory

Transcription

1 An introduction to Homotopy Type Theory Nicola Gambino University of Palermo Leicester, March 15th, 2013

2 Outline of the talk Part I: Type theory Part II: Homotopy type theory Part III: Voevodsky s Univalent Foundations

3 Part I: Type theory

4 Motivation for type theory Problem How can we write correct programs? Standard approach Write the program Verify its correctness via semantics Type-theoretic approach Write a correct-by-construction program

5 Verification via type-checking Idea Use types to classify syntactic expressions and write specifications Use type-checking to prevent mistakes Examples 3 : Nat cons([3, 4], [6, 2]) : List(Nat) [1, 7, 15, 34] : SortedList(Nat) [3, 1, 4, 8] : SortedList(Nat)

6 Type theories Goal Expressive type system Decidability of type checking Idea Powerful mechanism for defining recursive data types, e.g. Nat, List(A), Tree(A),... Dependent types, e.g. List n (A), is sorted(l).

7 Martin-Löf type theories Some forms of type: Empty, Unit, Bool, Nat, A B, A B, A + B, Id A (a, b), (Πx : A)B(x), (Σx : A)B(x),... We will only need the rules for identity types.

8 Identity types Formation rule A : type a : A b : A Id A (a, b) : type For example, if a : A then Id A (a, a) : type Introduction rule a : A refl(a) : Id A (a, a)

9 Elimination rule p : Id A (a, b) x: A, y : A, u: Id A (x, y) C(x, y, u): type x: A c(x): C(x, x, refl(x)) J(a, b, p, c): C(a, b, p) Idea [x: A] a = b C(x, x) C(a, b) Similar to Lawvere s treatment of equality in categorical logic.

10 Computation rule a: A x: A, y : A, u: Id A (x, y) C(x, y, u): type x: A c(x): C(x, x, refl(x)) J(a, a, refl(a), c) = c(a): C(a, a, refl(a)) Idea [x: A] a: A a = a C(x, x) C(a, a) a: A C(a, a)

11 Part II: Homotopy type theory

12 Semantics of type theories Problems Set-theoretical semantics validates also: p : Id A (a, b) a = b : A p : Id A (a, b) p = refl(a) : Id A (a, b) which makes type-checking undecidable. It is difficult to reason within type theories without good models.

13 Dictionary Type theory A : type a : A x : A B(x) : type x : A, y : A Id A (x, y) Inductive types Homotopy theory A space a A B A fibration A [0,1] A A Homotopy-initial algebras

14 Some results Theorem (Awodey and Warren). The rules for identity types admit an interpretation in every category equipped with a weak factorisation system. Theorem (Gambino and Garner). The deduction rules for identity types determine a weak factorisation system on the syntactic category of a Martin-Löf type theory. Theorem (Garner and van den Berg, Lumsdaine). Every type of Martin-Löf type theory determines a weak ω-groupoid. Theorem (Voevodsky). Martin-Löf type theories have models in the category of simplicial sets that do not validate the reflection rule.

15 Part III: Voevodsky s Univalent Foundations

16 While working on the completion of the Bloch-Kato conjecture I have thought a lot about what to do next. Eventually I became convinced that the most interesting and important directions in current mathematics are the ones related to the transition into a new era which will be characterized by the widespread use of automated tools for proof construction and verification. V. Voevodsky (2010)

17 Univalent Foundations Overview Use the dictionary of Homotopy Type Theory to introduce topological notions in type theory Exploit these notions to develop mathematics in type theory Formalise the development in Coq/Agda.

18 Contractibility Definition. A type X is contractible if the type is inhabited. iscontr(x) = def (Σx 0 : X)(Πx : X)Id X (x 0, x) Idea Existence and uniqueness Note X contractible X Unit X contractible Id X (x, y) contractible for all x, y : X

19 The hierarchy of h-levels Definition. A type X has level 0 if it is contractible. A type X has level n + 1 if for all x, y : X, the type Id X (x, y) has level n Terminology. Types of h-level 1 are called h-propositions (logic) Types of h-level 2 are called h-sets (algebra) Types of h-level 3 are called h-groupoids (category theory) Note. There is a +2 shift w.r.t. homotopy types.

20 Further developments Voevodsky s Univalence Axiom Calculations of fundamental groups of spheres Development of category theory...